Solving Quadratic Inequalities

Quadratic inequalities are mathematical statements of comparison in which a quadratic expression is set greater than or equal to another quadratic or lower degree expression.

For example `3x^2 + 5x + 6 ≥ x + 5` is a quadratic inequality because the highest degree on any term is 2.

Solving quadratic inequalities:

Move all terms to the left side

`3x^2 + 5x + 6  - x - 5 ≥ 0`
`3x^2 + 4x + 1 ≥ 0`

Find the zeros of the quadratic polynomial on the left side

`3x^2 + 4x + 1 = 0`
`3x^2 + 3x + x + 1 = 0`
`3x(x + 1) + 1(x + 1) = 0`
`(x + 1)(3x + 1) = 0`
`x + 1 = 0, 3x + 1 = 0`
`x = -1, x = -1/3`
If the quadratic expression doesn't factor, then find its zeros by the quadratic formula. If the quadratic expression doesn't have any zeros then evaluate it at any value of 'x' to see whether it is greater than or lesser than zero.

Make a number line and divide it into segments by the zeros


Take values in between the zeros and before and after the zeros and plug them in the quadratic polynomial to evaluate it.

`x = -2`
`3x^2 + 4x + 1 = 3(-2)^2 + 4(-2) + 1 = 5`

`x = -1/2`
`3x^2 + 4x + 1 = 3(-1/2)^2 + 4(-1/2) + 1 = 3/4 - 2 + 1 = 3/4 - 1 = -1/4`

`x = 0`
`3x^2 + 4x + 1 = 3(0)^2 + 4(0) + 1 = 1`

Wherever the quadratic polynomial evaluates negative, it is lesser than zero and wherever it evaluates positive it is greater than zero. Notice that the quadratic inequality has a greater than zero sign. So the regions of the number line where the quadratic polynomial evaluates positive are the answers. These are
  • Before `-1: (-∞, -1)`
  • After `-1/3: (-1/3, ∞)`
Now since the inequality symbol also includes the equal sign as it is "greater than or equal to (≥)", so the zeros are also taken up. To include the zeros -1 and -1/3, put square brackets near them:

So the answer is `(-∞, -1] U [-1/3, ∞)`

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